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Perhaps you are a contributor to the Stacks project and you would like to see your name spelled differently or you would like your name displayed in your native script. Although there are all kinds of technical difficulties with this, Pieter Belmans and I would like to try to do this. If you are interested please email your name as you would like to have it. For example try sending the info using Unicode or HTML escape characters.

If you want to leave a comment about a lemma, definition, remark, etc, please click through to the page of the lemma, definition, remark, etc and then leave a comment. Only leave a comment on a section if your comment is about the whole section or about text which is not in any environment, e.g., it is about the introduction to the section. This should be pretty rare.

If you have a longish comment involving mathematics and especially latex code, it is helpful to send a copy of your comment to the email address for the Stacks project (stacks.project..at..gmail.com). This will make it much faster for me to incorporate your suggested changes.

Technologically advanced people can use github and do a pull request.

Finally, if you don’t want to comment in public, then you can email the stacks project at the address mentioned above. Of course, I will be the one dealing with your email, since I am currently the maintainer of the project. So you could also decide to email to my personal email address. But if you do so, then there is a chance that I will overlook your email due to the large volume of incoming email in my personal email account. So I strongly prefer emails addressing issues with the Stacks project to be sent to the Stacks project email address.

One of the original goals of the Stacks project was to work through most of the “preliminary” material in the paper of Deligne and Mumford. Here I mean the material on algebraic stacks and on moduli stacks of curves, before one actually gets to the “interesting” part, namely, why the moduli stack of curves of a given genus is irreducible. This is now done. Currently the last theorem of the Stacks project is about how the moduli stack Mgbar is a proper and smooth Deligne-Mumford stack over Z for g >= 2.

Enjoy!

PS: I will make an effort to write more frequently here about what is going on with the Stacks project. In particular, I should write about the very successful Stacks project workshop which we just had, about what is next in line to be put in the Stacks project, about the wonderful people who help out with the Stacks project, and about how we’d like more people to help Pieter Belmans to code up parts of the new Stacks project web site!

Thanks for all your comments on the Stacks project. I especially enjoy the comments pointing out actual mathematics errors and even more those that calmly explain what went wrong and how to fix it. But all comments are good. We now have more than 250 people who have contributed a bit (and some contributed a lot). Thanks to all.

Remember this challenge? Probably not. But wait, don’t click! Namely, I will do something more general in this post.

Suppose we have a ring A and a contravariant functor F on (Sch/A) with the following properties:

F satisfies the sheaf property for fpqc coverings

the value of F on a scheme is either a singleton or empty

for every quasi-compact scheme T/A such that F(T) is nonempty, there is an ideal I of A such that F(Spec(A/I)) is nonempty and such that T —> Spec(A) factors through Spec(A/I).

Example: A = k[x, y] for a field k and F(T) is nonempty if and only if the generic point of Spec(A) is not in the image of T —> Spec(A). Here F is not a representable functor.

I’d like to add some conditions that guarantee that F is representable by a closed subscheme of Spec(A). Here is what I just came up with; I think it is obviously correct and the right thing to do. If A is Noetherian we add the following two conditions

If s_1, s_2, s_3, … is an infinite sequence of points of Spec(A) such that F(s_i) is nonempty and s is a limit point of the sequence, then F(s) is nonempty.

I leave it as an exercise to show that 1 — 5 imply F is representable in the desired manner. If A is not Noetherian, then somehow these should still be enough although maybe you need to replace the natural numbers by a bigger directed set.

Why am I excited by this observation? It is because I want to apply this to the situation of the other blog post mentioned above: X is an algebraic space of finite presentation over A, u : H —> G is a map between quasi-coherent O_X-modules. We assume G is flat over A, of finite presentation, and universally pure relative to A (this is a technical condition which is satisfied if the support of G is proper over A). The functor F is defined by F(T) is nonempty if and only if the base change u_T of u is zero.

Properties 1, 2, 3 hold for F and are easy to prove. The proof of property 4 still doesn’t use purity of G relative to A (I think because we already have 3 it follows from an argument using generic freeness, but I also have an argument using \’etale localization). The key is to prove property 5.

To see 5 is true, I argue as follows. Suppose that the base change u_∞ to A_∞ is nonzero. Choose a weakly associated point ξ of the image of u_∞. This is also a weakly associated point of G_∞. The image t’ of ξ in Spec(A_∞) specializes to a point t in V(I_1) = Spec(A_1) because I_1 is contained in the radical of A_∞. Because G is universally pure relative to A, there is a specialization θ of ξ which lies over t (indeed this is the definition of being pure relative to the base). Then since u_∞ is zero at θ (in a suitable \’etale neighbourhood Edit: Argh… I just discovered this doesn’t work!) it is zero at ξ, a contradiction.

Enjoy!

PS: A finitely presented module G on X flat and pure over A is universally pure relative to A. However, this is harder to prove than the above and it is easy to see that support proper over A implies universal purity.

Next summer sometime in July or August Wei Ho, Pieter Belmans, and I will organize a workshop on algebraic geometry and the Stacks project. Please go to the workshop webpage for more information (not a lot there yet) and to pre-register.